Filtered algebra

Summary

A filtered algebra is an associative algebra with an increasing chain of subspaces $F_0 \subseteq F_1 \subseteq \cdots$ that exhausts it and is compatible with multiplication, $F_i F_j \subseteq F_{i+j}$. It generalizes a graded ring, where elements have an exact degree, to a setting where elements only have a degree "at most $i$". Each filtered algebra has an associated graded algebra, which is often simpler, and many properties pass from the graded algebra back to the filtered one. The universal enveloping algebra of a Lie algebra and the Weyl algebra are the classic examples.